Optical metrology techniques generally referred to as scatterometry offer the potential to characterize parameters of a workpiece during a manufacturing process. In practice, light is directed onto a periodic grating formed in a workpiece and spectra of reflected light is measured and analyzed to characterize the grating parameters. Characterization parameters may include critical dimensions (CD), sidewall angle (SWA), feature height (HT), etc. which effect a material's reflectivity and refractive index. Characterization of the grating may thereby characterize the workpiece as well as a manufacturing process employed in the formation of the grating and the workpiece.
For the past several years, a rigorous couple wave approach (RCWA) and similar algorithms have been widely used for the study and design of diffraction structures. In the RCWA approach, the profiles of periodic structures are approximated by a given number of sufficiently thin planar grating slabs. Specifically, RCWA involves three main operations, namely, the Fourier expansion of the field inside the grating, calculation of the eigenvalues and eigenvectors of a constant coefficient matrix that characterizes the diffracted signal, and solution of a linear system deduced from the boundary matching conditions. RCWA divides the problem into three distinct spatial regions: 1) the ambient region supporting the incident plane wave field and a summation over all reflected diffracted orders, 2) the grating structure and underlying non-patterned layers in which the wave field is treated as a superposition of modes associated with each diffracted order, and 3) the substrate containing the transmitted wave field.
The accuracy of the RCWA solution depends, in part, on the number of terms retained in the space-harmonic expansion of the wave fields, with conservation of energy being satisfied in general. The number of terms retained is a function of the number of diffraction orders considered during the calculations. Efficient generation of a simulated diffraction signal for a given hypothetical profile involves selection of the optimal set of diffraction orders at each wavelength for both transverse-magnetic (TM) and/or transverse-electric (TE) components of the diffraction signal. Mathematically, the more diffraction orders selected, the more accurate the simulations. However, the higher the number of diffraction orders, the more computation is required for calculating the simulated diffraction signal. Moreover, the computation time is a nonlinear function of the number of orders used.